Heuristics Based on Unit Propagation for Satisability Problems. Chu Min Li & Anbulagan

Transcription

1 Heuristics Based on Unit Propagation for Satisability Problems Chu Min Li & Anbulagan LaRIA, Univ. de Picardie Jules Verne, 33, Rue St. Leu, Amiens Cedex, France fax: (33) , fcli@laria.u-picardie.fr, Anbulagan@utc.frg Abstract The paper studies new unit propagation based heuristics for Davis-Putnam-Loveland (DPL) procedure. These are the novel combinations of unit propagation and the usual "Maximum Occurrences in clauses of Minimum Size" heuristics. Based on the experimental evaluations of dierent alternatives a new simple unit propagation based heuristic is put forward. This compares favorably with the heuristics employed in the current state-of-the-art DPL implementations (C-SAT, Tableau, POSIT). 1 Introduction Consider a propositional formula F in Conjunctive Normal Form (CNF) on a set of Boolean variables fx 1 x 2 ::: x n g, the satisability (SAT) problem consists in testing whether clauses in F can all be satised by some consistent assignment of truth values (1 or 0) to the variables. If it is the case, F is said satisable otherwise, F is said unsatisable. If each clause exactly contains r literals, the subproblem is called r-sat problem. SAT problem is fundamental in many elds of computer science, electrical engineering and mathematics. It is the rst NP-Complete problem [Cook, 1971] with 3-SAT as the smallest NP-Complete subproblem. The Davis-Putnam-Loveland procedure (DPL) [Davis et al., 1962] is a well known complete method to solve SAT problems, roughly sketched in Figure 1. DPL procedure essentially constructs a binary search tree, each recursive call constituting a node of the tree. Recall that all leaves (except eventually one for a satisable problem) of a search tree represent a dead end where an empty clause is found. The branching variables are generally selected to allow to reach as early as possible a dead end, i.e. to minimize the length of the current path in the search tree. The most popular SAT heuristic actually is Mom's heuristic, which involves branching next on the variable having Maximum Occurrences in clauses of Minimum Size [Dubois et al., 1993 Freeman, 1995 Pretolani, 1993 Crawford and Auton, 1996 Jeroslow andwang, 1990]. Intuitively these variables allow to well exploit the power of unit propagation and to augment the chance to reach an empty clause. Recently another heuristic based on Unit Propagation (UP heuristic) has proven useful and allows to exploit yet more the power of unit propagation [Freeman, 1995 Crawford and Auton, 1996 Li, 1996]. Given a variable x, a UP heuristic examines x by respectively adding the unit clause x and x to F and independently makes two unit propagations. The real eect of the unit propagations is then used to weigh x. procedure DPL(F) Begin if F is empty, return "satisfiable" F:=UnitPropagation(F ) If F contains an empty clause, return "unsatisfiable". /* branching rule */ select a variable x in F according to a heuristic H, if the calling of DPL(F [fxg) returns "satisfiable" then return "satisfiable", otherwise return the result of calling DPL(F [fxg). End. procedure UnitPropagation(F) Begin While there is no empty clause and a unit clause l exists in F, assign a truth value to the variable contained inl to satisfy l and simplify F. Return F. End. Figure 1: DPL Procedure However, since examining a variable by two unit propagations is time consuming, two major problems remain open: should one examine all free variables by unit propagation at everynodeofasearch tree? if not, what are the variables to be examined at a search tree node?

2 In this paper we try to experimentally solve these two problems to obtain an optimal exploitation of UP heuristic. We dene a predicate at a search tree node whose denotational semantics is the set of variables to be examined at a search tree node, i.e. x is to be examined if and only if (x) is true. By appropriately changing,we experimentally analyse the behaviour of dierent UP heuristics. We write 12 DPL procedures which are dierent only in and run these procedures on a very large sample of hard random 3-SAT problems. We begin in section 2 by describing the 12 DPL procedures and summarizing the experimental results on these programs. In section 3 we compare a pure UP heuristic and a pure Mom's heuristic and show the superiority of UP heuristics. In section 4 we study dierent restrictions of UP heuristics. In section 5 we discuss the related work and compare the best DPL procedure in our experimentation with three state-of-the-art DPL procedures. Section 6 concludes the paper. 2 UP Heuristics Driven by Let diff(f 1 F 2 ) be a function which gives the number of clauses of minimum size in F 1 but not in F 2,we show a generic branching rule in Figure 2, where the equation dening H(x) is suggested in [Freeman, 1995] and the weight 5 for uniformizing clauses of dierent length is empirically optimal. For each free variable x such that (x) is true do let F 0 and F 00 be two copies of F Begin F 0 := UnitPropagation(F 0 [fxg) F 00 := UnitPropagation(F 00 [fxg) If both F 0 and F 00 contain an empty clause then return "F is unsatisfiable". If F 0 contains an empty clause then x := 0, F := F 00 else if F 00 contains an empty clause then x := 1, F := F 0 If neither F 0 nor F 00 contains an empty clause then let w(x) denote the weight of x w(x) :=diff(f 0 F) and w(x) :=diff(f 00 F) End If all variables examined above are valued or (x) is false for every x then For each free variable x in F do let r i be the length of the clause C i w(x) := x2c i 5 ;r i and w(x) := x2c i 5 ;r i For each variable x do H(x) :=w(x) w(x) w(x)+w(x) Branching on the free variable x such that H(x) is the greatest. Figure 2: A Generic Branching Rule Driven by The essential reason to use UP heuristics instead of Mom's one is that Mom's heuristic may not maximize the eectiveness of unit propagation, because it only takes binary clauses (if any) into account to weigh a variable, although some extensions try to also take longer clauses into account with exponentially smaller weights (e.g. 5 ternary clauses are counted as 1 binary clauses). A UP heuristic allows to take all clauses containing a variable and their relations into account in a very eective way to weigh the variable. As a secondary eect, it allows to detect the so-called failed literals in F which when satised falsify F in a single unit propagation. However since examining a variable by two unit propagations is time consuming, it is natural to try to restrict the variables to be examined. For this purpose we use predicate dened at a search tree node. The success of Mom's heuristic suggests that the larger the number of binary occurrences of a variable is, the higher its probability of being a good branching variable is, implying that if one should restrict UP heuristics by means of, he should restrict UP heuristics to those variables having a sucient number of binary occurrences. For this reason, all restrictions on studied in this paper are dened according to the number of binary occurrences of a variable, so that the resulted UP heuristics rely on combinations of unit propagation and Mom's heuristics inclusion relation Figure 3: predicate hierarchy by inclusion relation. ij(x) is true i x has i binary occurrences of which at least j negative and j positive. The rst predicate in our experimentation is called a and has empty denotational semantics, the resulted branching rule using a pure Mom's heuristic, and the second is called 0 whose denotational semantics is the set of all free variables, the resulted UP heuristic is in its pure form and plays its full role: a : a (x) is false for every free variable x 0 : 0 (x) is true for every free variable x. Between a and 0, there are many other possible predicates. Figure 3 denes 8 predicates constituting a hierarchy by inclusion relation of their denotational semantics is dened to be 31, but for nodes under a xed depth of a search tree, it is dened to be

3 41. Let T be a constant, the last predicate is named z and is dened to be the rst of the three predicates 41, 31 and 0 (in this order) which has at least T variables in its denotational semantics. Each predicate results in a DPL procedure, a, 0, 3141 and z respectively giving Sata, Sat0, Sat 3141 and Satz, and ij giving Sat ij. These programs are dierent only in predicate, except Sat0 which need not count the occurrences of variables. We run the 12 programs (compiled using gcc with optimization) on a PC with a 133 Mhz Pentium CPU under Linux operating system on a very large sample of random 3-SAT problems generated by using the method of Mitchell et al.[mitchell et al., 1992]. Given a set V of n Boolean variables fx 1 x 2 ::: x n g,we randomly generate m clauses of length 3. Each clause is produced by randomly choosing 3 variables from V and negating each with probability 0.5. Empirically, when the ratio m=n is near 4.25 for a 3-SAT formula F, F is unsatisable with a probability 0.5 and is the most dicult to solve. We vary n from 140 variables to 340 variables incrementing by 20, for each n the ratio clauses-to-variables (m=n) is set to 4.0, 4.1, 4.2, 4.25, 4.3, 4.4, 4.5. At each ratio and by each program, if n <280 then 1000 problems are solved, if 280 n 300 then 500 problems are solved, if n = 320 then 300 problems are solved, and if n = 340 then 100 problems are solved. A problem is solved successively by all the 12 DPL procedures before another to ensure the same environment to all programs. Due to the lack of space, we only present the experimental results for the ratio m=n = 4.25 in Figures 4, 5, and 6, where the DPL procedures corresponding to the curves are listed in the same order from top to bottom. The experimental results on the other ratios give exactly the same conclusions. 3 A Pure UP Heuristic Versus a Pure Mom's Heuristics: Sat0 vs Sata Sat0 systematically examines all the variables by unit propagation at all nodes, using a pure UP heuristic, while Sata does not examine any variable so and employs a pure Mom's heuristic. One might believe that Sat0 would be simply too slow, but it is not the case. Sat0 is much faster than Sata. In fact from Figures 4 and 5, all DPL procedures using a UP heuristic in our experimentation are substantially better than Sata in terms of search tree size and real run time. Note that Mom's heuristic used in Sata is similar to the so-called two-sided Jeroslow-Wang rule [Hooker and Vinay, 1995], with the only dierence that a clause of length i is counted as 5 clauses of length i+1 instead of 2. Our experiments suggest that 5 is better than 2. 5 is also similar to the exponential factors in C-SAT [Dubois et al., 1993] where 5.71 ternary clauses are counted as 1 binary clause. search tree size (number of nodes) x 106 sata sat42 sat40 sat41 sat3141 sat30 sat31 sat21 sat20 satz sat10 sat nb. of variables Figure 4: Mean search tree size of each program as a function of n for hard random 3-SAT problems at the ratio m=n =4.25 time in sec sata sat42 sat10 sat40 sat20 sat30 sat0 sat21 sat41 sat3141 sat31 satz nb. of variables Figure 5: Mean run time of each program as a function of n for hard random 3-SAT problems at the ratio m=n = 4.25 Sat0 actually is slower than ve other programs based on balanced restrictions of variables to be examined by unit propagation, but not substantially so (except Satz). The surprisingly good performance of Sat0 conrms the power of UP heuristics for selecting the next branching variable and suggests that its eect for detecting failed literals is only secondary. 4 Restricted UP Heuristics Figure 6 illustrates the number of variables examined by dierent restricted UP heuristics at a node.

4 average number of variables examined per search tree node sat10 sat20 sat21 sat30 sat31 sat3141 sat40 sat41 sat search tree depth Figure 6: Average number of variables examined at a search tree node in a given depth when solving hard random 3-SAT problems of 300 variables and 1275 clauses (500 problems are solved) for 9 programs 4.1 Restriction by total number of binary occurrences of a variable Four programs Sat 10, Sat 20, Sat 30 and Sat 40 realize this type of restrictions. While a classical Mom's heuristic selects the next branching variable having maximum binary occurrences, the restricted UP heuristics examine a set of variables having more binary occurrences than others, including the variable having maximum binary occurrences. From Figure 4, it is clear that the more variables are examined, the smaller the search tree size is. 4.2 Balanced restriction by total number of binary occurrences of a variable Four programs Sat 21, Sat 31, Sat 41 and Sat 42 realize this type of restrictions. The predicates require that a variable occurs both positively and negatively in binary clauses to balance the search tree. We compare the duet Sat i0 and Sat i1 (i=2, 3, 4) and observe that Sat i1 examines strictly fewer variables than Sat i0 and is faster than it in spite of a slightly larger search tree. In particular, Sat 41 and Sat 40 examine almost the same number of variables (see Figure 6), but the balanced restriction gives a faster DPL procedure. We pay special attention to 31 and 41 since they seem to be the best balanced restrictions. 4.3 Dynamic restriction as a function of search tree depth Sat 3141 realizes this restriction. A general observation when solving 3-SAT problems using a DPL procedure is that there are more and more binary clauses when descending from the search tree root and the denotational semantics of a predicate such as 31 becomes larger and larger. Furthermore, the nodes are more numerous near the leaves and the branching variables play a less important role there. It appeared that one could restrict more the variables to be examined by unit propagation near the leaves without important loss on the search tree size so as to obtain some gain in terms of real run time. POSIT's UP heuristic (called BCP-based heuristic) [Freeman, 1995] realizes this idea: under the level 9 of a search tree, at most 10 variables are examined by unit propagation. Sat 3141 uses 31 from the top of a search tree, but under the depth empirically xed to n 4=70, it uses 41, where n is the numberofvariables in the initial input 3-SAT problem. Note that if n 160, n 4=70 9, so Sat 3141 generally strengthens the restriction later than POSIT. From Figures 4 and 5 Sat 3141 is not better than Sat 31, although it makes many fewer unit propagations to examine variables (see Figure 6), suggesting that the search tree depth is rather irrelevant to the restriction of UP heuristics. 4.4 Dynamic restriction by number of variables to be examined The relatively poor performance of Sat 42 seems due to the small number of variables examined at each node (see Figure 6), though these variables have many binary occurrences. A careful analysis shows that even Sat 31, the best one up to now, examines few or no variables at some nodes, especially near the root where there are few binary clauses, although these nodes are more determinant for the nal search tree size. z is then introduced to ensure that at least T variables are examined at each node, T being empirically xed to 10. Near the root, all free variables are examined to exploit the full power of UP heuristic. As soon as the number of variables occurring both negatively and positively in binary clauses and having at least 4 (3) binary occurrences is larger than T, only these variables are examined to select the next branching variable. 5 Related Work C-SAT [Dubois et al., 1993] examines some variables by unit propagations (called local processing) near the bottom of a search tree to rapidly detect failed literals there. Pretolani also uses a similar approach (called pruning method) based on hypergraphs in H2R [Pretolani, 1993]. But the local processing and the pruning method as are respectively presented in [Dubois et al., 1993] and [Pretolani, 1993] do not contribute to the heuristic to select the next branching variable. We nd the rst eective exploitation of UP heuristic in POSIT [Freeman, 1995]

5 and Tableau [Crawford and Auton, 1996] which use a similar idea as in C-SAT to determine the variables to be examined at a node by unit propagation: x is to be examined i x is among the k most weighted variables by a Mom's heuristic. The main dierence of Satz with Tableau and POSIT is that Satz does not specify a upper bound k of the number of variables to be examined at a node by unit propagation. Instead, Satz species a lower bound. In fact, Satz examines many more variables by an optimal combination of unit propagation and Mom's heuristics. Given the depth of a node, Table 1 illustrates the average numberofvariables examined (#examined vars) at the node by Satz, with the depth of the root being 0. In order to compare with C-SAT, Tableau and POSIT we also give the theoretical value of k C (for C-SAT), k T (for Tableau) and k P (for POSIT) at the node, respectively according to the denitions of k in [Dubois et al., 1993 Crawford and Auton, 1996 Freeman, 1995]. depth #free vars #examined vars k C k T k P or or or or or or or or or or or or or or or 3 Table 1: Average number of variables examined in Satz at a node in a given depth when solving a hard random 3- SAT problem of 300 variables and 1275 clauses (500 problems are solved) compared with theoretical value of k in C-SAT, Tableau and POSIT It is clear that Satz examines many more variables at each node than any of C-SAT, Tableau or POSIT. Near the root, Satz examines all free variables. Elsewhere Satz examines a sucient number (T )ofvariables. We compare C-SAT, Tableau, POSIT and Satz on a large sample of hard random 3-SAT problems on a SUN Sparc 20 workstation with a 125 MHz CPU. The 3-SAT problems are generated from 3 sets of n variables and m clauses at the ratio m=n =4:25, n steping from 300 variables to 400 variables by 50. We use an executable of C-SAT dated July The versionoftableau used here is called 3tab and is the same used for the experimentation presented in [Crawford and Auton, 1996]. POSIT is compiled using the provided make command on the SUN Sparc 20 workstation from the sources named posit-1:0:tar:gz 1.Table 2 shows the performances of the 4 DPL procedures on problems of 300, 350, and 400 variables, where time standing for the real mean run time is reported by the unix command /usr/bin/time and t size standing for search tree size (number of nodes) is reported (or computed from number of branches reported) by the DPL procedures. 300 vars 350 vars 400 vars 300 problems 250 problems 100 problems System time t size time t size time t size C-SAT Tableau POSIT Satz Table 2: Mean run time (in second) and mean search tree size of C-SAT, Tableau, POSIT and Satz on ratio m=n=4.25 Table 2 shows that Satz is faster than the above cited versions of C-SAT, Tableau and POSIT, Satz's search tree size is the smallest, and Satz's run time and search tree size grow more slowly. Table 3 shows the gain of Satz compared with the cited version of C-SAT, Tableau and POSIT at the ratio m=n=4.25. Each item is computed from Table 2 using the following equation: gain =(value(system)=value(satz) ; 1) 100% where value is real mean run time or real mean search tree size and system is C-SAT, Tableau or POSIT. From Table 3, it is clear that the gain of Satz grows with the size of the input formula. 300 vars 350 vars 400 vars 300 problems 250 problems 100 problems System time t size time t size time t size C-SAT 126% 51% 152% 58% 216% 77% Tableau 132% 31% 175% 45% 276% 66% POSIT 68% 89% 133% 130% 198% 200% Table 3: The gain of Satz vs. C-SAT, Tableau and POSIT in terms of run time and search tree size on the ratio m=n=4.25 computed from Table 2 The central strategy of Satz is to try to reach an empty clause as early as possible. Further along the line, we make two relatively small resolvents-driven improvements in Satz. The rst improvement is the preprocessing of the input formula by adding some resolvents of length 3, The second improvement consists in rening yet more the heuristic H in the nodes where all free variables are examined by unit propagation. Refer to Figure 2, when z is equal to 0 we dene w(x) as the number of resolvents the newly produced binary clauses would result in in F 0 by a single step of resolution. w(x) is similarly dened. 1 publicly available via anonymous ftp to ftp.cis.upenn.edu in pub/freeman/ directory

6 Satz improved in this way solves many real-world or structured SAT problems where previous heuristics were not successful. For example, Table 4 shows the performance of the 4 DPL procedures on the well-known Beijing challenging problems 2, where a problem that can not be solved in less than 2 hours is marked by "> 7200" and the version of Tableau is called ntab 3. It is clear that Satz is much more ecient and solves many more problems in less than two hours. Problem Satz C-SAT Posit ntab 2bitadd 10 > 7200 > 7200 > 7200 > bitadd > > bitadd > bitcomp bitmax bitadd 31 > 7200 > 7200 > 7200 > bitadd > 7200 > 7200 > blocks blocksb > blocks 1542 > 7200 > 7200 > 7200 e0ddr2-10-by > 7200 > 7200 > 7200 e0ddr2-10-by > enddr2-10-by-5-1 > 7200 > 7200 > 7200 > 7200 enddr2-10-by > 7200 > ewddr2-10-by > > 7200 ewddr2-10-by > 7200 > Table 4: Run time (in sec.) of Beijing challenging problems 6 Conclusion We found that UP heuristic is substantially better than Mom's one even in its pure form realized by 0 where all free variables are examined at all nodes. In its restricted forms based on combinations of unit propagation and Mom's heuristics, the more variables are examined, the smaller the search tree is, conrming the advantages of UP heuristic, but too many unit propagations slow the execution. The combinations realized by 41 and 31 represent good compromises. A dynamic restriction such as 3141 which strengthens the restriction under a xed depth of a search tree fails to work better than the static restriction 31.We design the dynamic restriction along another line: z ensures that at least T candidates are examined by unit propagation at every node of a search tree by successively using 41, 31 and 0, giving the very ecient and very simple DPL procedure called Satz. Satz is favorably compared with several current stateof-the-art DPL implementations (C-SAT, Tableau and POSIT) on a large sample of hard random 3-SAT problems and the recent Beijing SAT benchmarks. The good performance of Satz on the structured or real-world SAT problems shows that UP heuristic can tackle new problems or problem domains where Mom's heuristics were not successful and enhances the belief that if a DPL procedure is ecient for random SAT problems, it should be also ecient for a lot of structured ones. Acknowledgments We thank Olivier Dubois, James M. Crawford and Jon W. Freeman for kindly providing us their DPL procedures and anonymous referees for their comments which helped improve this paper. References [Chvatal and Szemeredi, 1988] V. Chvatal and E. Szemeredi. Many Hard Examples for Resolution. Journal of ACM, 35(4):759{768, October [Cook, 1971] S. A. Cook. The Complexity of Theorem Proving Procedures. In 3rd ACM Symp. on Theory of Computing, pages , Ohio, [Crawford and Auton, 1996] J. M. Crawford and L. D. Auton. Experimental Results on the Crossover Point in Random 3-SAT. Articial Intelligence, 81, [Davis et al., 1962] M. Davis, G. Logemann, and D. Loveland. A machine program for theorem proving. Communication of ACM, 5(7): , July [Dubois et al., 1993] Olivier Dubois, P. Andre, Y. Boufkhad and Jacques Carlier. SAT versus UNSAT. Second DIMACS Challenge: Cliques, Coloring and Satisability, Rutgers University, NJ, [Freeman, 1995] Jon W. Freeman. Improvements to Propositional Satisability Search Algorithms. Ph.D. thesis, Department of computer and Information science, Univ. of Pennsylvania, Philadelphia, PA, [Hooker and Vinay, 1995] J. N. Hooker and V. Vinay. Branching Rules for Satisability. Journal of Automated Reasoning, 15: , [Jeroslow andwang, 1990] R. Jeroslow and J. Wang. Solving Propositional Satisability Problems. Annals of Mathematics and AI, 1: , [Li, 1996] ChuMin LI. Exploiting Yet More the Power of Unit Clause Propagation to Solve 3-SAT Problem. In ECAI'96 Workshop on Advances in Propositional Deduction, pages 11-16, Budapest, Hungary, [Mitchell et al., 1992] D. Mitchell, B. Selman, H. Levesque. Hard and Easy Distributions of SAT Problems. In AAAI'92, pages 459{465, San Jose, CA, [Pretolani, 1993] Daniele Pretolani. Satisability and hypergraphs. Ph.D. thesis, Dipartimento di Informatica, Universita di Pisa, available from 3 available from